An Invitation to Fourier Analysis and Distribution Theory
Description
This book provides a rigorous introduction to Fourier analysis and the theory of distributions without presupposing an advanced background in functional analysis or measure-theoretic integration.
The guiding principle throughout is to present the material in an elementary and direct manner without sacrificing mathematical precision or depth. After introducing Fourier series and their fundamental properties in the first chapter, the main ideas of Fourier analysis are developed first for non-periodic functions on Euclidean space, where many concepts can be presented more transparently, and are subsequently transferred to the periodic setting. Likewise, distribution theory is initially developed in the framework of tempered distributions, which allows for a simpler exposition and is particularly well suited to applications in partial differential equations, including those discussed in Chapter 7. Distributions on open subsets of Euclidean space are then introduced via localization, emphasizing their inherently local character. The final chapter presents an elementary proof of the Schwartz kernel theorem based on expansions in Hermite functions, from which the tensor product of distributions is obtained as an immediate consequence. Each chapter concludes with a collection of exercises ranging from routine applications to more challenging problems.
More details
Person
Detlef Müller is Professor of Mathematics at the Kiel University, Germany. An distinguished expert in analysis and partial differential equations, he has published over 100 research articles and served on the editorial board of prestigious journals. His honors include an invited lecture at the International Congress of Mathematicians in Berlin (1998) and a Fellowship of the AMS.
Content
1 The Basic Idea of Fourier Analysis: Expansion of Periodic Functions into Trigonometric Series.- 2 Fourier Transform and Convolution on R??.- 3 Fourier Series and the Poisson Summation Formula.- 4 Tempered Distributions.- 5 Distributions in Open Subsets of R??.- 6 Distributions with Compact Support.- 7 Fundamental Solutions.- 8 On the Regularity Theory of Linear Partial Differential Equations: The Singular Support and Hypoellipticity.- 9 The Schwartz Kernel Theorem, and the Tensor Product of Distributions.- Appendix A: The Baire Category Theorem and the Banach-Steinhaus Theorem*.